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P54 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破

单位正方形内的最低星偏差 · n = 37

在单位正方形里放 n 个采样点。从原点量起、边平行于坐标轴的每一块矩形，占了多少面积，就该分到多少比例的点；你的分数，是所有这类矩形里最大的那个偏差。把它压到最低。

子题n = 37
目标最小化 最大误差 D*

严格定义

  • 容器单位正方形，左下角是原点 (0, 0)，右上角是 (1, 1)
  • 提交恰好 n 个点的坐标，十进制小数，最多九位；两点不得重合
  • 矩形被比较的矩形是半开的 [0, u) × [0, v)，右上角可落在正方形内任何位置；恰好压在上边或右边上的点算在外面
  • 目标让所有矩形中最大的偏差 D* 尽可能小。上确界在提交坐标的网格上取得，以整数精确计分
放大来摆，然后提交 ↗
1y0
0x1
已验证构造虚线框占 62.0% 的面积，照这个比例应该有 22.9 个点，实际有 37 个，差 0.3798，这就是分数

帮助理解

一个比喻：采样预算

把正方形当成一帧要渲染的画面，这 n 个点就是你全部的采样预算。哪块矩形分到的点比面积应得的多，是预算浪费在同一处；少了，是那块的细节被漏掉。

哪里有优化空间

格点和随机撒点都会在某些矩形上系统性偏置；低偏差构造（Hammersley、van der Corput）压得低得多。但对每个具体的 n，没人知道还能压到哪里。

前沿在哪里

n ≤ 21 的最优解已由 Clément、Doerr、Klamroth 与 Paquete 在 2025 年证明（Proc. Amer. Math. Soc. Ser. B 12: 78–90），所以本站从 n = 22 起；再往上，每一个 n 都是开放的。

当前第一名

0.379846603359478

最大误差 D*

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

容器是边长 1 的正方形：左下角是原点 (0, 0)，右上角是 (1, 1)。被比较的矩形永远从原点量起，右上角可以落在正方形里的任何位置，所以一共有无穷多块矩形要同时满足，而不是某几块。坐标写成小数，例如 "0.25"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。榜上的数字是 D*，也就是所有矩形里最大的那个误差；分数越小越好。

当前第一名的答案

{
  "points": [
    [
      "0.010810811",
      "0.010810811"
    ],
    [
      "0.032432432",
      "0.410810811"
    ],
    [
      "0.054054054",
      "0.210810811"
    ],
    [
      "0.075675676",
      "0.610810811"
    ],
    [
      "0.097297297",
      "0.110810811"
    ],
    [
      "0.118918919",
      "0.510810811"
    ],
    [
      "0.140540541",
      "0.310810811"
    ],
    [
      "0.162162162",
      "0.710810811"
    ],
    [
      "0.183783784",
      "0.060810811"
    ],
    [
      "0.205405405",
      "0.460810811"
    ],
    [
      "0.227027027",
      "0.260810811"
    ],
    [
      "0.248648649",
      "0.660810811"
    ],
    [
      "0.27027027",
      "0.160810811"
    ],
    [
      "0.291891892",
      "0.560810811"
    ],
    [
      "0.313513514",
      "0.360810811"
    ],
    [
      "0.335135135",
      "0.760810811"
    ],
    [
      "0.356756757",
      "0.035810811"
    ],
    [
      "0.378378378",
      "0.435810811"
    ],
    [
      "0.4",
      "0.235810811"
    ],
    [
      "0.421621622",
      "0.635810811"
    ],
    [
      "0.443243243",
      "0.135810811"
    ],
    [
      "0.464864865",
      "0.535810811"
    ],
    [
      "0.486486486",
      "0.335810811"
    ],
    [
      "0.508108108",
      "0.735810811"
    ],
    [
      "0.52972973",
      "0.085810811"
    ],
    [
      "0.551351351",
      "0.485810811"
    ],
    [
      "0.572972973",
      "0.285810811"
    ],
    [
      "0.594594595",
      "0.685810811"
    ],
    [
      "0.616216216",
      "0.185810811"
    ],
    [
      "0.637837838",
      "0.585810811"
    ],
    [
      "0.659459459",
      "0.385810811"
    ],
    [
      "0.681081081",
      "0.785810811"
    ],
    [
      "0.702702703",
      "0.023310811"
    ],
    [
      "0.724324324",
      "0.423310811"
    ],
    [
      "0.745945946",
      "0.223310811"
    ],
    [
      "0.767567568",
      "0.623310811"
    ],
    [
      "0.789189189",
      "0.123310811"
    ]
  ]
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 37
}

当前第一名的答案

{
  "points": [
    [
      "0.010810811",
      "0.010810811"
    ],
    [
      "0.032432432",
      "0.410810811"
    ],
    [
      "0.054054054",
      "0.210810811"
    ],
    [
      "0.075675676",
      "0.610810811"
    ],
    [
      "0.097297297",
      "0.110810811"
    ],
    [
      "0.118918919",
      "0.510810811"
    ],
    [
      "0.140540541",
      "0.310810811"
    ],
    [
      "0.162162162",
      "0.710810811"
    ],
    [
      "0.183783784",
      "0.060810811"
    ],
    [
      "0.205405405",
      "0.460810811"
    ],
    [
      "0.227027027",
      "0.260810811"
    ],
    [
      "0.248648649",
      "0.660810811"
    ],
    [
      "0.27027027",
      "0.160810811"
    ],
    [
      "0.291891892",
      "0.560810811"
    ],
    [
      "0.313513514",
      "0.360810811"
    ],
    [
      "0.335135135",
      "0.760810811"
    ],
    [
      "0.356756757",
      "0.035810811"
    ],
    [
      "0.378378378",
      "0.435810811"
    ],
    [
      "0.4",
      "0.235810811"
    ],
    [
      "0.421621622",
      "0.635810811"
    ],
    [
      "0.443243243",
      "0.135810811"
    ],
    [
      "0.464864865",
      "0.535810811"
    ],
    [
      "0.486486486",
      "0.335810811"
    ],
    [
      "0.508108108",
      "0.735810811"
    ],
    [
      "0.52972973",
      "0.085810811"
    ],
    [
      "0.551351351",
      "0.485810811"
    ],
    [
      "0.572972973",
      "0.285810811"
    ],
    [
      "0.594594595",
      "0.685810811"
    ],
    [
      "0.616216216",
      "0.185810811"
    ],
    [
      "0.637837838",
      "0.585810811"
    ],
    [
      "0.659459459",
      "0.385810811"
    ],
    [
      "0.681081081",
      "0.785810811"
    ],
    [
      "0.702702703",
      "0.023310811"
    ],
    [
      "0.724324324",
      "0.423310811"
    ],
    [
      "0.745945946",
      "0.223310811"
    ],
    [
      "0.767567568",
      "0.623310811"
    ],
    [
      "0.789189189",
      "0.123310811"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。榜上的数字是 D*，也就是所有矩形里最大的那个误差；分数越小越好。 · 验证器 v1.0.0